1. Field of the Invention
The invention relates to a method for solving deconvolution problems where it is desired to reconstruct a signal over a time range or another variable of interest. It finds particular application in determining a flux from a source which is spaced from a detector, such as determining flux from a biological cell or layer of cells, and will be described with particular reference thereto. It will be appreciated, however, that the invention is equally applicable to a wide range of deconvolution problems, curve fitting problems, spectral shape recognition and analysis, calibration, and the like, where the independent variable may be time or spatial or other coordinates.
2. Discussion of the Art
Transport at the cellular level is an essential element for sustaining life. Anomalies in cellular transport have been associated with a host of conditions, ranging from Cystic Fibrosis to Multidrug Resistance (MDR) in cancer cells. In order to treat such life threatening conditions, it is desirable to develop a qualitative and quantitative understanding of the underlying transport mechanisms and their anomalies. Cellular release is a very common mode of transport used by cells to make adjustments to changes in their environment in order to maintain homeostasis and to respond to external stimuli. Hence, understanding of the type and quantities of species released by a particular cell type can assist in understanding the associated biological processes taking place.
Transport at a cell cluster, or at single cells, may involve release, cellular efflux, uptake, mass transport in the extracellular medium, or any combination of these processes. The quantity characterizing these processes is generally called a flux, or a flux density, and is expressed in units of moles (or weight) per unit area per unit time. The ability to provide accurate plots of flux over time has particular application in the study of cellular transport mechanisms.
There are many applications when it is desirable to reconstruct flux values from signals detected some distance from the source of the flux. For example, in studying efflux of a chemical, such as an ion or drug, from a monolayer of biological cells, such as human or other animal cells, the monolayer is covered by a liquid, such as a cell medium or buffer. A sensor, such as an electrode system, is placed in contact with the liquid, at a finite distance from the monolayer. The sensor measures a concentration of the drug or ion (hereinafter chemical species) in the adjacent liquid, rather than the actual flux of the drug or ion secreted from the monolayer. This is because the measured concentration depends on not only the efflux at the monolayer surface, but also on the diffusion of the chemical species, and the distance of the sensor from the monolayer. The chemical species passes into the liquid and diffuses through the liquid toward the sensor over time, at a rate governed by the diffusion constant. While concentration measurements can provide some useful information, they can give only an indirect indication of the flux at the cells. This is because the flux across the plasma membrane in either direction induces a secondary form of transport in the extracellular medium or space, whose own, relatively slow dynamics can largely affect what the sensor actually senses. It is therefore desirable to reproduce, as closely as possible, the flux values at the cells, which provide a more accurate picture of what is going on at the cell level.
Current measurement schemes for directly obtaining flux values at a cell's or cell cluster's surface are limited to the study of some charged particles (a few inorganic ions) and are restricted in the information they provide due to the lack of sufficient sensitivity. Hence, in most contexts, flux values have to be indirectly reconstructed from concentration changes monitored at some fixed distance from the cell site.
For determination of ionic fluxes, the concentration is generally measured by placing an ion selective electrode at a fixed distance from the surface of the cell monolayer. The ion selective electrode produces a voltage change corresponding to the change in the concentration of the specific ion that it is designed to measure. For example, a potassium ion selective electrode shows a potential change corresponding to a change in concentration of potassium ions. Irrespective of the measurement scheme used, the concentration that is measured is not a direct indication of the secretion flux. This is because the concentration is affected by other parameters, such as the distance of sensor from the monolayer, the diffusivity of the ions in the solution, and the dynamics of mass transport between the source and the sensor.
To derive flux from measured concentrations, the mass transport from cells to the sensor is computationally “undone”. This leads to deconvolution, which is known to be a problematic mathematical operation.
Mathematical expressions for the relationships between concentration and flux have been developed. By assuming mass transport to be planar and one dimensional, the measured concentration can be related to the secretion flux by the differential equation:                                           ∂            C                                ∂            t                          =                  D          ⁢                                    ∂                                                                   2                                ⁢                C                                                    ∂                              z                2                                                                        (        1        )            where t is the time (seconds);                z is the distance from the monolayer (cm);        F is the flux at the cell monolayer (e.g., in (mmol cm−2 s−1), which can be expressed as a function of time: F(t) represents the flux at the surface at a time corresponding to the time at the which the emitted flux was generated;        C is the measured concentration (e.g., in mmol cm−3), which can be expressed as a function of distance from the monolayer and time: C(z,t); and        D is the diffusion constant.        
C(z,0)=0 is the initial condition. i.e., at time t=0 the concentration at a distance z from the monolayer is zero. Other initial conditions may exist, however.
The assumption of planar mass transport is generally valid where sufficient time has passed before any measurement such that the diffusion zones of the different point sources (cells within the monolayer) fully overlap. The electrode is placed far enough from the monolayer of cells so that it “sees” the sources as one planar source. There is no diffusion in the negative z direction and hence the assumption of one dimensional diffusion is also valid.
The boundary condition is represented by the expression:                               F          ⁡                      (            t            )                          =                              -                          D              A                                ⁢                                    (                                                ∂                  C                                                  ∂                  z                                            )                                      z              =              0                                                          (        2        )            
where F is as defined above and A is the surface area of the monolayer.
The diffusion problem can be expressed by the equation:                               δ                      z            ,            t                          =                              1                                          π                ⁢                                                                   ⁢                Dt                                              ⁢                      exp            ⁡                          (                                                -                                      z                    2                                                                    4                  ⁢                                                                           ⁢                  Dt                                            )                                                          (        3        )            
The delta function in this equation translates the effect of secretion at a cell surface, (−F(t) (mmol cm−2 s−1) into an amount of material in a unit volume, Cz(t) (mmol cm−3) over a period of time and hence has a unit of cm−1. The general solution to the diffusion expression is provided by the expression:                                           C            z                    ⁡                      (            t            )                          =                              ∫            0            t                    ⁢                                    F              ⁡                              (                                  t                  ′                                )                                      ⁢                          δ              ⁡                              (                                  t                  -                                      t                    ′                                                  )                                      ⁢                                                   ⁢                          ⅆ                              t                ′                                                                        (        4        )                            where F (t′) represents the flux at the surface at a time t′, corresponding to the time at which the emitted flux corresponding to the measured concentration was generated.        
The measured concentration C(t) is a convolution of the flux F(t) with the delta function (Eqn. 4). Hence the convolution of flux with the delta function is often described as a “forward problem,” the solution of which is the measured concentration. Deriving the flux from the measured concentration is called the “inverse problem.” Various methods have been used to solve this “inverse” problem (obtaining F(t) knowing C(t)), often termed “deconvolution”. In general, a plot of measured concentrations over time is compared with a plot of concentrations obtained from a proposed flux over time. The currently available methods, however, do not usually achieve a close fit with the actual flux plot.
Inverse Fourier Transform and Error Minimization techniques are the most commonly used tools for solving such deconvolution problems. In the Inverse Fourier Transform method, Fourier transform techniques are used to generate the flux plots. The Fourier transform utilizes the following expression in its calculations:F(t)=F−1[F(Cz(t))]/F[δ(t)])  (5)                where F denotes the Fourier operation and F−1 denotes its inverse.        
Fourier transform methods have been widely used for a number of deconvolution applications, in part because the complicated integration in the time domain is converted into simple multiplication in the Fourier domain. However the Fourier transform theory makes assumptions which are not valid in all applications. For example, the method assumes that the function (flux in the present case) to be determined is a periodic one. In many instances, such as the flux from cells, this assumption is invalid. Moreover, it assumes that negative flux values are possible, which again does not generally correspond with reality. Moreover, for cell mass transport, the resulting flux is not an instantaneous result of whatever is its cause, but rather occurs after a time delay. This again contributes to errors in the reconstructed flux data, such as negative values for the reconstructed flux. Thus, due to artifacts introduced to the deconvolution based on these and other assumptions, the proposed flux does not accurately correspond with the actual flux. This tends to get even worse after repeating the procedure several times. The resulting plots often have negative flux values or sudden and unrealistic drops or increases in flux. More importantly, the computed flux will not reproduce the measured concentration plot when inserted into the forward problem.
In another method, error minimization, based on a least squares errors (LSQ) technique, is used to solve the inverse problem. In the LSQ method, error is defined asCr(t)=[Cr(t1)Cr(t2) . . . ];Cs(t)=[Cs(t1)Cs(t2) . . . ]  (6) Error=√(Cr(t1)−Cs(t1))2+(Cr(t2)—Cs(t2 ))2+ . . .
Where subscripts r and s correspond to real (actual) and assumed starting concentrations, respectively.
Error minimization techniques are aimed at optimizing square errors. Error minimization is performed using minF(t) [error (Cr(t), Cs(t)], where                Cr(t)=real or actual concentration (mmol cm−3);        Fs(t)=starting flux (mmol cm−2s−1) and        Cs(t)=concentration corresponding to the starting flux (mmol cm−3)        
Cr(t) is the concentration that is measured and hence is referred to as the measured or actual concentration, even though artifacts in the measuring device may lead to less than actual values being generated. To apply the least squares technique, a flux Fs(t) is assumed and the concentration Cs(t) corresponding to this flux is calculated using Equation 4. The error between Cs(t) and Cr(t) is calculated using Equation 6. The Flux value is then changed so that the error between Cs(t) and Cr(t) is decreased. The solution goes through numerous iterations, with the expectation that each time, the proposed flux plot will more closely correspond with the actual flux plot. This is tested by the difference between the actual concentration plot and the one predicted from the flux.
The method uses directional stepping in which each element of a flux vector is stepped, keeping the others constant. Based on these operations, a new flux vector is determined. The concentration corresponding to this new flux vector is then calculated. The error between the real concentration and the concentration obtained by stepping the elements of the flux vector is then calculated. This gives a directional error vector, which is the partial derivative of error with respect to individual elements of the flux vector.
This technique tends to introduce noise, unrealistic shapes in the flux and concentration plots, and instability into the results.
The present invention provides a new and improved optimization method which overcomes the above-referenced problems and others.